In this paper, the authors study a kind of boundary value problems for functional differential equations with nonlinear boundary conditions〖JB({〗εx″(t)=f(t,x(t),x(t-τ(t)),Tx(t),x′(t),ε),t∈(0,1),x(t)=φ(t,ε),t∈−τ,0,h(x(1),x′(1),ε)=A(ε),[JB)]where ε>0 is a small parameter, τ(t)≥τ\-0>0,τ=\%\{max\}\%[DD(X]t∈0,1[DD)]τ(t)<1,Tx(t)=ψ(t)+∫\+t\-0k(t,x)x(s)ds is a type of Volterra map. By using the t heory of differential inequality, we prove the existence of the solution and uniforml y valid asymptotic expansions of the solution is given as well.
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